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Theorem inindif 29192
Description: See inundif 4023. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
inindif ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Proof of Theorem inindif
StepHypRef Expression
1 inss2 3817 . . . 4 (𝐴𝐶) ⊆ 𝐶
21orci 405 . . 3 ((𝐴𝐶) ⊆ 𝐶𝐴𝐶)
3 inss 3825 . . 3 (((𝐴𝐶) ⊆ 𝐶𝐴𝐶) → ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶)
42, 3ax-mp 5 . 2 ((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶
5 inssdif0 3926 . 2 (((𝐴𝐶) ∩ 𝐴) ⊆ 𝐶 ↔ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
64, 5mpbi 220 1 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
Colors of variables: wff setvar class
Syntax hints:  wo 383   = wceq 1480  cdif 3557  cin 3559  wss 3560  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3897
This theorem is referenced by:  resf1o  29339  gsummptres  29561  measunl  30052  carsgclctun  30156  probdif  30255
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